


1. 

An object of mass 450g moves in
a circular path of radius 1.5m. 


It completes 2.5 revolutions per second. 


Calculate 

a) 
the angular velocity 

b) 
the linear speed of the body 

c) 
the magnitude of the centripetal force needed to maintain this
motion 

d) 
the work done by this force during 10 revolutions. 



2. 

A mass of 15kg is moving in a
circular path at the end of a metal rod 4m
long. 


The axis of rotation passes through the other end of
the rod and the plane of the motion is horizontal. 


The maximum tension that the rod can tolerate is
5×10^{4}N. 

a) 
Draw a diagram showing the force(s) acting on the mass. 


Ignore the force of gravity and assume there are no
frictional forces. 

b) 
Indicate clearly on the diagram the direction in which the mass
will move if the rod breaks. 

c) 
Calculate the maximum linear speed with which the mass can move
without breaking the rod. 

d) 
Calculate the maximum rotational frequency corresponding to this
maximum speed. 



3. 

Now consider a situation similar to the one described in the
previous question but this time we have a rod
8m long with a mass of 15kg
on each end. 


The axis of rotation now passes through the middle of
the rod. 


What is the maximum rotational frequency in this case, assuming
the rod can still take a maximum of 5×10^{4}N
tension. 



4. 

A 200g mass is moving in a
circular path on the end of a light metal rod 0.5m
long. 


The axis of rotation passes through the other end of
the rod and the plane of the motion is vertical. 


The rotational frequency is 1.2s^{1}. 

a) 
Calculate the angular velocity. 

b) 
Draw two diagrams showing the positions of the mass when the
tension in the rod is i) maximum and ii) minimum. 


Label the forces acting on the mass and explain briefly why the
tension varies. 

c) 
Calculate the magnitudes of the maximum and minimum tensions. 

d) 
Calculate the angular velocity at which the minimum tension is
zero. 



5. 

A body of mass 2kg is attached
to a string 1m long and moves in a
horizontal circle of radius 50cm, as
shown below. 











This arrangement is often called a "conical pendulum". 

a) 
Calculate the magnitude of the tension in the string. 

b) 
Calculate the time period of the motion (the time to complete
one revolution). 



6. 
a) 
At what angle should a road surface be banked in order that a
vehicle can go round a bend of radius 55m
at a speed of 40kmh^{1}? 

b) 
Suppose that a vehicle attempts to go round this bend at 100kmh^{1}. 


If the coefficient of friction between the wheels of the vehicle
and the road surface is 0.25, and the mass of the vehicle is 1.5tons
(1500kg), will the vehicle skid or
not? 


Show your calculations. 



7. 

A body, of mass m, is moving in a circular path of radius r 


The centripetal force acting on it is F_{c} 


Show that the rotational frequency of the motion is given by 


